Fractional Hamiltonian systems with locally defined potentials

A. B. Benhassine

Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 1, pp. 81-90

Résumé

We study solutions of the nonperiodic fractional Hamiltonian systems $$ -{}_tD^{\alpha}_{\infty}({}_{-\infty} D_{t}^{\alpha}x(t))-L(t)x(t)+ \nabla W(t,x(t))=0,\quad x\in H^\alpha(\mathbb{R},\mathbb{R}^N), $$ where $\alpha\in(1/2,1]$, $t\in\mathbb R$, $L(t)\in C(\mathbb R,\mathbb R^{N^2})$, and ${}_{-\infty}D^{\alpha}_{t}$ and ${}_tD^{\alpha}_{\infty}$ are the respective left and right Liouville–Weyl fractional derivatives of order $\alpha$ on the whole axis $\mathbb R$. Using a new symmetric mountain pass theorem established by Kajikia, we prove the existence of infinitely many solutions for this system in the case where the matrix $L(t)$ is not necessarily coercive nor uniformly positive definite and $W(t,x)$ is defined only locally near the coordinate origin $x=0$. The proved theorems significantly generalize and improve previously obtained results. We also give several illustrative examples.

Détail

Keywords: fractional Hamiltonian system, critical point theory, symmetric mountain pass theorem.